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What is meant is actually not the ''delta function'' but the so-called Kronecker delta tensor. This tensor is written as the greek letter delta and has two indices, so it can be represented as a matrix. This matrix is just the identity matrix. If I denote the Kronecker delta by ''delta'', then the following holds:

delta_{mu,nu}X_{nu} = X_{mu}

I.e. if you apply delta to a vector you get the same vector back, simply because it is the identity matrix which has a 1 on the diagonal. In terms of indices, delta_{mu, nu} is zero unless mu and nu are equal, in which case it is 1. In the above equation where you sum over the repeated index nu, all terms are zero except when nu hits mu and therefore you get X_{mu}.

The question in the problem deals with the metric tensor which relates so-called covariant and contravariant tensors. The distinction between these two types of tensors is important in General Relativity, but the notation is also used in electromagnetism and particle physics. Vectors, tensors etc. are useful because these allow you to write down the laws of physics in a way that doesn't depend on the particular choice of your coordinate system. This is called covariance.

Now, you'll remember from linear algebra that on a vector space you can define an inner product. The inner of two vectors is always a real number. But from relativity you'll remember that the standard inner product of two vectors is not an invariant. Instead for two four-vectors X and Y, the following quantity is the same for all inertial observers: